# Is "underlying" the right word?

I am describing a mathematical model, where the probability density function of a variable is made up of two contributions, two distributions. Mathematically we would say that f(x) = g1(x) + g2(x).

Now: in the text I am writing something like this:

"the distribution f(x) is the sum of two underlying distributions, g1(x) and g2(x). [...] in order to estimate the parameters of the underlying distributions we use a parametric approach...."

I am trying to use "underlying" in the sense of "lying under".

What do you think? Is underlying the right word or should I address those distributions in a different way (how)?

This is a perfectly legitimate use of the word, and I recall encountering it and using it myself in writing about statistics. It sounds fine to at least one native English speaker with a degree in mathematics.

• As a non-mathematician it made sense to me, too. I wonder, though, can operand be used as a modifier in this sort of situation? Would something like "operand distributions" make sense to a mathematician?
– user13141
Commented Oct 15, 2011 at 11:33
• It's been a looong time, but I don't think I ever encountered that phrase. "Underlying distribution" is a fairly common idiom. Commented Oct 15, 2011 at 13:33
• I think everybody would be confused by "operand distributions", but "component distributions" would work fine. Commented Oct 15, 2011 at 15:55
• I agree - I was just about to suggest "component distributions" when I saw this comment. In some ways, this is a better choice of words. Commented Oct 15, 2011 at 18:05
• @Peter, James: surely component would strongly suggest that g1 and g2 are coming from orthogonal subspaces? It’s not quite clear to me if that’s the case in the OP’s situation.
– PLL
Commented Oct 20, 2011 at 18:20

I'd leave out 'underlying', 'in order', and some articles, and correct the mathematical terminology, because the sum of two probability distributions is not a probability distribution, but their average is. That is, I'd rephrase "the distribution f(x) is the sum of two underlying distributions, g1(x) and g2(x)... in order to estimate the parameters of the underlying distributions we use a parametric approach..." as "Distribution f(x) is the average of distributions g1(x) and g2(x)... to estimate the parameters of g1 and g2, we use a parametric approach...".

Update (Response to 2 comments)

I see use of 'underlying' as pompously verbose, and also somewhat misleading. While one could say, for example, "7 is the sum of the underlying numbers 2 and 5," it is far more direct to say "7 equals 2 plus 5." The word 'underlying' contributes nothing to this simple example, and contributes relatively little to the question's example.

Given the further information that the sum is a weighted sum of multiple distributions, I have less objection to use of 'underlying', although still see it as verbose, and for the second instance of it might substitute 'basis' if appropriate. For example: "Distribution f(x) is a weighted sum of distributions g_1(x) ... g_k(x)... we use a parametric approach to characterize the g_i basis functions...".

• If you suggest leaving out "underlying" (why? what do you think is wrong with it?), to actually answer the question, you should suggest a word to replace it. "Underlying" (or a similar replacement word) adds context and helps the reader understand what is going on here. I don't see any reason to get rid of it. The phrase "weighted average of two underlying distributions" works fine as well, and is clearer than just "of two distributions". Commented Oct 15, 2011 at 16:03
• Thank for your contribution, jwpat7. However, my example was just a big "simplification" of my case, so I cannot use the word "average", since the resulting distribution is a weighted sum of the "underlying" distributions Commented Oct 15, 2011 at 16:08
• I would assume, from "this distribution is the sum of two underlying distribution", that in a probability model for how this distribution arises there are contributions from two different types of processes; this is similar to "this disease has two underlying causes". Yes, you can leave out "underlying" in either sentence, but in my opinion this results in a slight loss of clarity. Further, if you introduce the word "underlying", you can later use it to distinguish between the first distribution and the underlying distributions. Calling them all just "distributions" could be confusing. Commented Oct 15, 2011 at 16:55